Teaching Mathematics: What, When and Why

An in-depth examination of mathematics education, topic by topic

What is the O in BODMAS?

Hint: It’s not “order”.

The herd seems to be following an internet meme that O stands for “Order”. And what does that mean? “Power” or “Index” we are told. Other equally silly suggestions for O are “Operations” and “Other”.

Once upon a time, the 1950s in my case, BODMAS was taught long before the introduction of indices. It was to specify precedence when combining the five basic arithmetic operations. Yes five: addition, subtraction, multiplication, division and of. In those days mathematics in school was closer to that used in common parlance; it included the operation “of”. As in “three quarters of twelve”. We quickly learned that of could by replaced by multiplication; so in the example “1/2 of 10 + 4” we perform the of before the addition. One web site
http://www.notjustsums.co.uk/2015/02/what-does-o-stand-for-in-bodmas.html
nearly gets it. It mentions of but claims that applies to powers, as in “powers of”!

By the way, most people seem to agree that multiplication and division have equal precedence, but this is a problem. For example, (10\div 2)\times 10 gives a different result from 10\div (2 \times 10). The standard solution is that runs of \times and \div are treated from left to right. This seems to always give the same result as giving division precedence over multiplication. Can someone prove or disprove that?

Now throw in the of operator. In everyday speech “ten divided by half of ten” means 10\div(1/2 \times 10) which is 2. But treating of as just multiplication gives 10\div 1/2 \times 10 = 200. So of requires higher precedence than division (and multiplication). We cannot just treat of as the same as multiply, it demands parentheses. The teachers who first coined BODMAS knew where to place the O.

Before we get too precious about these rules, note that any salt-worthy mathematician would insert parentheses in any expression where the precedence is not clear.

But how did the world forget the meaning of O? If you look at the current Victorian syllabus, the idea that “of” occurs as a mathematical operation is missing. Another casualty of New Maths? If you treat mathematics as an abstract system, divorced from its history and common usage, then there is no need for of when we have \times. So when students get to Year 11 and are incapable of everyday applications, we shunt them into a numeracy course. Good luck with that.

PS: I’ve just noticed, when the operator of is applied to integers it becomes ambiguous. So “three of seven” could mean “three lots of seven”, 3\times 7, or “three out of seven”, 3\div 7. All the more reason to discuss it in elementary school.

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21 responses to “What is the O in BODMAS?”

  1. tom Avatar
    tom

    In the version sent to me as an email, the latex formulae are wrongly placed. But going directly to the site it seems correct. Is anyone else getting a corrupt version?

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  2. Aaron Avatar
    Aaron

    Very interesting post. I actually did not know about BODMAS and would have dismissed the title as quibbling about unimportant things if I had not been aware of the quality of your posts. In terms of the format in the email, my email also had the latex formulas floating in the middle of the text (what you describe as corrupted).

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    1. tom Avatar
      tom

      Thanks Aaron for your kind words. I’m guessing that subscribers get an email advising them of the post, and that may be “corrupt”. But if you go directly to the site

      https://teaching-maths.org/2024/06/10/what-is-the-o-in-bodmas/

      then it appears “correct”? A similar thing happens to me when I access Marty’s site

      https://mail.google.com/mail/u/0/#inbox/FMfcgzQVwxCxJwRVHxZsltxQqNnGcWhT

      Yes the O in BODMAS is a minor matter, which I would normally ignore. But I have come to see it as symptomatic of a deeper problem. Hence this current rant.

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  3. Glen Wheeler Avatar
    Glen Wheeler

    Hi Tom,

    I think you’re right.

    I wonder if function composition confused anyone as well?

    cheers

    Glen

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    1. tom Avatar
      tom

      Had not thought of that; can you discuss examples?

      In my day it was called “function of a function”. But again there is a deeper issue. New Math decreed that a function can only have one value. This makes for more difficult issues and much fretting about “domain” and “range”. When a learner reaches Complex Variable theory, then multivalued functions become necessary, unless you want to destroy the beautiful Riemann theory. And when she reaches Point Set Topology, inverse functions are unashamedly multi-valued.

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  4. Glen Wheeler Avatar
    Glen Wheeler

    Good point.

    In high school, they might talk about log “of” e^x, or say sin “of” arccos “of” x (then proceed to simplify it based on triangles and Pythagoras). It’s fairly innocent in terms of order of operations, but I just wonder if that’s what some people may think the “O” in BODMAS stands for (function of a function as you said).

    Cheers

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    1. tom Avatar
      tom

      In olden times we might write log sin 2pi where the order of operations was assumed to be log (sin (2pi)) rather than (log (sin 2))pi. I guess the new practice in schools of always including parentheses is safer.

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      1. tom Avatar
        tom

        Damned WordPress editor will not allow me to correct the silly latex typesetting!

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  5. Tom’s Story of O – BAD MATHEMATICS Avatar
    Tom’s Story of O – BAD MATHEMATICS

    […] Peachey has written a new post on his blog: What is the O in BODMAS? As I have indicated, I don’t really care. Nonetheless, […]

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  6. Terry Mills Avatar
    Terry Mills

    Thanks Tom. I’ll tell a story rather than comment on the purpose of your note.

    A couple of years ago, I was teaching some Year 11 students and I discovered that they did not have a good understanding of order of operations. So I tried to teach them – before an upcoming examination where I anticipated that such a question might arise. I started with a problem – I can’t remember it. But most of the class got it wrong. They insisted that they were using BODMAS – but they were not using it correctly. So I asked them to use their calculators to solve the problem. They got the same answer that I had. When I pointed this out to them, their reply was that, obviously, the calculator was not using BODMAS. Fortunately I was later told by the person who set the examination that students could use their calculators in the exam.

    Best wishes, Terry

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    1. tom Avatar
      tom

      Hi Terry. Still teaching?

      Do you recall the error in their BODMAS?

      Since the order of precedence is a matter of convention, then I guess the result of a calculator carries some weight.

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  7. marty Avatar
    marty

    Hi, Tom. Interesting post. I hadn’t realised the “of” had been forgotten. I wonder if it might still appear somewhat in primary school, where kids are introduced to BODMAS, and where they definitely get plenty of “of” in companion with a fraction multiplier.

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    1. tom Avatar
      tom

      Yes “of” still gets a guernsey but not it seems in the formal BODMAS where it’s precedence is given.

      So far nobody has commented on my claim that the everyday application of mathematics has been downgraded in elementary school. This is based on my (very limited) tutoring experience at this level . Of course we no longer need to calculate the pay of someone who works 5 days at 1 pound, 7 shillings and threepence halfpenny per day , plus 4 1/2 hours overtime at time and a half.

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      1. marty Avatar
        marty

        What do you mean by “everyday application of mathematics”? Primary school maths teaching is infested with “real world” garbage. NAPLAN is pretty much nothing else. Are simply objecting that students can’t do arithmetic?

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      2. tom Avatar
        tom

        I agree that the major problem is that students cannot do arithmetic.

        My memory of primary school mathematics was that it was useful for me doing the billing for our family business. Can you give examples of the “garbage”.

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      3. marty Avatar
        marty

        Oh, Jesus. The easiest example is any NAPLAN test. The arithmetic is trivial, and the insane over-emphasis is on the words. But it’s much worse in regular classes. The word “arithmetic” no longer exists. What matters is “numeracy”, which means pointless and dull as a dirt applications of arithmetic to the real world. 95% application and 5% trivial arithmetic, or calculator. A massive time sucker.

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  8. JJ Avatar
    JJ

    Thanks – fascinating – I wondered if I had misremembered but your post clarifies it. The old fashioned ways seem to work better is my conclusion as I watch my poor students struggle with algebra without the times tables and mental arithmetic that make it so much easier.

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  9. Tim Avatar
    Tim

    We were taught Brackets Of Division Multiplication Addition Subtraction in primary school. The Of seemed redundant to me.

    We were taught Brackets Orders Division Multiplication Addition Subtraction in high school. This seemed like an improvement, except the O still seems redundant.

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    1. tom Avatar
      tom

      Looks like I didn’t convince you. My point was that although “of” is computed using multiplication, its priority order in natural speech is different from $latex\times$. And the designers of BODMAS were alive to this subtlety and so included “of” in the scheme. That, plus my usual rant about how the modern syllabus is too abstract at school level.

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  10. MathsIsKing Avatar
    MathsIsKing

    can I stand in front of the class and claim I have no idea about bodmas?

    but that they should fix inside of brackets, powers, then multipl and div from left to right, then the same with addition and subtr.

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  11. tom Avatar
    tom

    I’m not sure what you mean. Would you like to enlarge, using standard English?

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